Constant mean curvature surfaces in Minkowski 3-space via loop - PowerPoint PPT Presentation

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Constant mean curvature surfaces in Minkowski 3-space via loop groups David Brander Now: Department of Mathematics Kobe University (From August 2008: Danish Technical University) Geometry, Integrability and Quantization - Varna 2008

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Outline CMC Surfaces in Euclidean Space

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Outline CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space The loop group construction

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Constant Mean Curvature Surfaces in Euclidean 3-space • Soap films are CMC surfaces. • Air pressure on both sides of surface the same ↔ mean curvature H = 0, minimal surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Minimal Surfaces: H = 0 • Gauss map of a minimal surface is holomorphic . Figure: Costa’s surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Minimal Surfaces: H = 0 • Gauss map of a minimal surface is holomorphic . • Weierstrass representation: pair of holomorphic functions ↔ minimal surface Figure: Costa’s surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space CMC H � = 0 Surfaces • Gauss map is a harmonic (not holomorphic) map into S 2 = SU ( 2 ) / K , K = { diagonal matrices } . Figure: A constant non-zero mean curvature surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space CMC H � = 0 Surfaces • Gauss map is a harmonic (not holomorphic) map into S 2 = SU ( 2 ) / K , K = { diagonal matrices } . • Loop group frame F λ . Figure: A constant non-zero mean curvature surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space CMC H � = 0 Surfaces • Gauss map is a harmonic (not holomorphic) map into S 2 = SU ( 2 ) / K , K = { diagonal matrices } . • Loop group frame F λ . • Can recover f from the loop group map F λ via a simple formula. Figure: A constant non-zero mean curvature surface

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods • Λ G C = { γ : S 1 → G C | γ smooth } • F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods • Λ G C = { γ : S 1 → G C | γ smooth } • F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a • Example: flat surfaces in S 3 .   ω λβ λθ F − 1  = a 0 + a 1 λ, − λβ t λ d F λ = 0 0  − λθ t 0 0 order ( 0 , 1 ) .

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a 1. AKS theory: 2. KDPW Method: 3. Dressing:

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a 1. AKS theory: Constructs order ( 0 , b ) maps, b > 0, by solving ODE’s. Related to inverse scattering. 2. KDPW Method: 3. Dressing:

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a 1. AKS theory: Constructs order ( 0 , b ) maps, b > 0, by solving ODE’s. Related to inverse scattering. 2. KDPW Method: Constructs order ( a , b ) maps, a < 0 < b , from a pair of ( a , 0 ) and ( 0 , b ) maps. 3. Dressing:

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Loop Group Methods F λ : M → Λ G C is of connection order ( a , b ) if b F − 1 � a i λ i . λ d F λ = a 1. AKS theory: Constructs order ( 0 , b ) maps, b > 0, by solving ODE’s. Related to inverse scattering. 2. KDPW Method: Constructs order ( a , b ) maps, a < 0 < b , from a pair of ( a , 0 ) and ( 0 , b ) maps. 3. Dressing: Any kind of connection order ( a , b ) maps. Produces families of new solutions from a given solution.

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Krichever-Dorfmeister-Pedit-Wu (KDPW) Method • Need Birkhoff factorization : Λ G C “ = ” Λ + G C · Λ − G C , where Λ ± G C consists of loops which extend holomorphically to D and ˆ C \ D resp.

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Krichever-Dorfmeister-Pedit-Wu (KDPW) Method • Need Birkhoff factorization : Λ G C “ = ” Λ + G C · Λ − G C , where Λ ± G C consists of loops which extend holomorphically to D and ˆ C \ D resp. • If F λ is of order ( a , b ) , a < 0 < b , decompose F = F + G − = F − G + .

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Krichever-Dorfmeister-Pedit-Wu (KDPW) Method • Need Birkhoff factorization : Λ G C “ = ” Λ + G C · Λ − G C , where Λ ± G C consists of loops which extend holomorphically to D and ˆ C \ D resp. • If F λ is of order ( a , b ) , a < 0 < b , decompose F = F + G − = F − G + . • Then F + is of order ( 0 , b ) and F − is of order ( a , 0 ) :

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Krichever-Dorfmeister-Pedit-Wu (KDPW) Method • Need Birkhoff factorization : Λ G C “ = ” Λ + G C · Λ − G C , where Λ ± G C consists of loops which extend holomorphically to D and ˆ C \ D resp. • If F λ is of order ( a , b ) , a < 0 < b , decompose F = F + G − = F − G + . • Then F + is of order ( 0 , b ) and F − is of order ( a , 0 ) : F − 1 G − ( F − 1 d F ) G − 1 − + G − d G − 1 + d F + = − b � a i λ i ) G − 1 − + G − d G − 1 = G − ( − a = .

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Krichever-Dorfmeister-Pedit-Wu (KDPW) Method • Need Birkhoff factorization : Λ G C “ = ” Λ + G C · Λ − G C , where Λ ± G C consists of loops which extend holomorphically to D and ˆ C \ D resp. • If F λ is of order ( a , b ) , a < 0 < b , decompose F = F + G − = F − G + . • Then F + is of order ( 0 , b ) and F − is of order ( a , 0 ) : F − 1 G − ( F − 1 d F ) G − 1 − + G − d G − 1 + d F + = − b � a i λ i ) G − 1 − + G − d G − 1 = G − ( − a c 0 + ... + c b λ b . =

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space KDPW Method • Conversely, given order ( 0 , b ) and ( a , 0 ) maps, F + and F − , we can construct an order ( a , b ) map F . • After a normalization, both directions unique: � F + � F ← → F − ( 0 , b ) ( a , b ) ( a , 0 )

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Specific Case Harmonic Maps into Symmetric Spaces • G / K symmetric space, K = G σ . • On Λ G C , define involution ˆ σ : (ˆ σγ )( λ ) := σ ( γ ( − λ )) .

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Specific Case Harmonic Maps into Symmetric Spaces • G / K symmetric space, K = G σ . • On Λ G C , define involution ˆ σ : σ ⊂ Λ G C σ ⊂ Λ G C . • Fixed point subgroup Λ G ˆ ˆ

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space • F λ ( z ) a connection order ( − 1 , 1 ) map, C → Λ G ˆ σ . • KDPW:

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space • F λ ( z ) a connection order ( − 1 , 1 ) map, C → Λ G ˆ σ . • KDPW: F ↔ { F + , F − } • In this case, F + determined by F − , so F ↔ F −

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space • F λ ( z ) a connection order ( − 1 , 1 ) map, C → Λ G ˆ σ . • KDPW: F ↔ F − • Fix λ ∈ S 1 : then F λ : C → G .

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space • F λ ( z ) a connection order ( − 1 , 1 ) map, C → Λ G ˆ σ . • KDPW: F ↔ F − • Fix λ ∈ S 1 : then F λ : C → G . • Fact: Projection of F , to G / K , is a harmonic map C → G / K if and only if F − is holomorphic in z :

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space • F λ ( z ) a connection order ( − 1 , 1 ) map, C → Λ G ˆ σ . • KDPW: F ↔ F − • Fix λ ∈ S 1 : then F λ : C → G . • Fact: Projection of F , to G / K , is a harmonic map C → G / K if and only if F − is holomorphic in z : order ( − 1 , 1 ) F ↔ F − order ( − 1 , − 1 ) harmonic holomorphic

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space “Weierstrass Representation” for CMC H � = 0 Surfaces • a ( z ) , b ( z ) arbitrary holomorphic. Set � 0 � a ( z ) λ − 1 d z . α = b ( z ) 0

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space “Weierstrass Representation” for CMC H � = 0 Surfaces • a ( z ) , b ( z ) arbitrary holomorphic. Set � 0 � a ( z ) λ − 1 d z . α = b ( z ) 0 • Automatically, d α + α ∧ α = 0. Integrate to get F − : Σ → Λ G , connection order ( − 1 , − 1 ) .